Commutator collecting process

In mathematical group theory, the commutator collecting process is a method for writing an element of a group as a product of generators and their higher commutators arranged in a certain order. The commutator collecting process was introduced by Philip Hall (1934). He called it a "collecting process" though it is also often called a "collection process".
Statement

The commutator collecting process is usually stated for free groups, as a similar theorem then holds for any group by writing it as a quotient of a free group.

Suppose F₁ is a free group on generators a₁, ..., a_m. Define the descending central series by putting

F_n+1 = [F_n, F₁]

The basic commutators are elements of F1 defined and ordered as follows.

The basic commutators of weight 1 are the generators a1, ..., a_m.
The basic commutators of weight w > 1 are the elements [x, y] where x and y are basic commutators whose weights sum to w, such that x > y and if x = [u, v] for basic commutators u and v then y ≥ v.

Commutators are ordered so that x > y if x has weight greater than that of y, and for commutators of any fixed weight some total ordering is chosen.

Then F_n/F_n+1 is a fnitely-generated free abelian group with a basis consisting of basic commutators of weight n.

Then any element of F can be written as

\( g=c_1^{n_1}c_2^{n_2}\cdots c_k^{n_k}c \)

where the ci are the basic commutators of weight at most m arranged in order, and c is a product of commutators of weight greater than m, and the ni are integers.

References

Hall, Marshall (1959), The theory of groups, Macmillan, MR 0103215
Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society 36: 29–95, doi:10.1112/plms/s2-36.1.29
Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, pp. 90–93, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050

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